Abstract

We introduce two proximal iterative algorithms with errors which converge strongly to the common solution of certain variational inequality problems for a finite family of pseudocontractive mappings and a finite family of monotone mappings. The strong convergence theorems are obtained under some mild conditions. Our theorems extend and unify some of the results that have been proposed for this class of nonlinear mappings.

1. Introduction

In many problems, for example, convex optimization, linear programming, monotone inclusions, elliptic differential equations, and variational inequalities, it is quite often to seek a proximal point of a given nonlinear problem. The proximal point algorithm is recognized as a powerful and successful algorithm in finding a common point of the fixed points of pseudocontractive mappings and the solutions of monotone mappings. Let be a closed convex subset of a real Hilbert space with inner product and norm . We recall that a mapping is called monotone if and only if

A mapping is called -inverse strongly monotone if there exists a positive real number such that

Obviously, the class of monotone mappings includes the class of the -inverse strongly monotone mappings. The class of monotone mappings is one of the most important classes of mappings among nonlinear mappings. The classical variational inequality problem is formulated as finding a point such that , for all . The set of solutions of variational inequality problems is denoted by .

A mapping is called pseudocontractive if, for all , we have

A mapping is called -strict pseudocontractive if there exists a constant such that

A mapping is called nonexpansive if

Clearly, the class of pseudocontractive mappings includes the class of strict pseudocontractive mappings and the class of nonexpansive mappings. We denote by the set of fixed points of ; that is, .

A mapping is called contractive with a contraction coefficient if there exists a constant such that

Recently viscosity approximation methods for finding fixed points of pseudocontractive mappings have received vast investigations because of their extensive applications in a variety of applied areas of partial differential equations, image recovery, and signal processing. In Hilbert spaces, many authors have studied the fixed-point problems of the nonexpansive mappings and monotone mappings by the viscosity approximation methods and obtained a series of good results; see [1–18] and the reference therein.

For finding an element of the set of fixed points of the nonexpansive mappings, Halpern [1] was the first to study the convergence of the scheme in 1967:

In 2000, Moudafi [2] introduced the viscosity approximation methods and proved the strong convergence of the following iterative algorithm under some suitable conditions:

Takahashi et al. [19, 20] introduced the following scheme and studied the weak and strong convergence theorems of the elements of , respectively, under different conditions:
where is a nonexpansive mapping and is an -inverse strong monotone operator. Recently, Zegeye and Shahzad [21] introduced the mappings as follows:

Very recently, Tang [22] introduced the following sequence and obtained the strong convergence theorems:
For other related results, see [11–13, 23–25]. On the other side, there are perturbations always occurring in the iterative processes because the manipulations are inaccurate. There is no doubt that researching the convergent problems of iterative methods with perturbation members is a significant job. Starting from any initial guess, , the proximal point algorithm generates a sequence according to the inclusion:
where is a maximal monotone operator and is a parameter. For solving the original problem of finding a solution to the inclusion , Rockafellar [23] introduced the following algorithm:
where is a sequence of errors. Rockafellar [23] obtained the weak convergence of the algorithm. Very recently Yao and Shahzad [24] proved that sequences generated from the method of resolvent are given by
where is a sequence in , the sequence is a small perturbation, and is a nonexpansive mapping.

The following is our concern now: Is it possible to construct a new sequence with general errors that converges strongly to a common element of fixed points of pseudocontractive mappings and the solution set of monotone mappings and converges strongly to the unique solution of certain variational inequality?

In this paper, motivated and inspired by the above results, we introduce two iterations with perturbations which converge strongly to a common element of the set of fixed points of a finite family of pseudocontractive mappings more general than nonexpansive mappings and the solution set of a finite family of monotone mappings more general than -inverse strongly monotone mappings or maximal monotone mappings. Our theorems presented in this paper improve and extend the corresponding results of Yao and Shahzad [24], Zegeye and Shahzad [21], and Tang [22] and some other results in this direction.

2. Preliminaries

In the sequel, we will use the following lemmas.

Lemma 1 (see [6]). Let be a sequence of nonnegative real numbers satisfying the following relation:
where is a sequence in (0,1) and is a real sequence such that (i);(ii) or .
Then .Let be a nonempty closed and convex subset of a real Hilbert space ; a mapping is called the metric projection if, for all , there exists a unique point in , denoted by such that
It is well known that is a nonexpansive mapping.

Lemma 2 (see [25]). Let be a nonempty closed and convex subset of a real smooth Hilbert space . Let ; then have the property as follows:

Lemma 3 (see [21]). Let be closed convex subset of Hilbert space . Let be a continuous monotone mapping, let be a continuous pseudocontractive mapping, and define the mappings and as follows: for , ,
Then the following hold:(i) and are single valued;(ii) and are firmly nonexpansive mappings; that is, , ;(iii), ;(iv) and are closed convex.

Lemma 4 (see [11]). Let and be bounded sequences in a Banach space and let be a sequence in which satisfies the following condition:
Suppose
Then .

Lemma 5 (see [22]). Let be a nonempty closed convex and bounded subset of a Hilbert space , and let be a finite family of nonexpansive mappings such that . Suppose that and . Then there exists nonexpansive mapping such that .

3. Main Results

Let be closed convex subset of Hilbert space . Let be a finite family of continuous monotone mappings, and let be a finite family of continuous pseudocontractive mappings. For the rest of this paper, and are mappings defined as follows: for , ,
By using Lemmas 2.3–2.6 in Zegeye and Shahzad [21], we have that the mappings and are well defined and they are nonexpansive and , are closed convex. Denote , .

Theorem 6. Let be a nonempty closed convex subset of uniformly smooth strictly convex real Hilbert space . Let be a finite family of continuous pseudocontractive mappings, let be a finite family of continuous monotone mappings such that , and let be contraction with a contraction coefficient . and are defined as (21) and (22), respectively. Let be a sequence generated by :
where , , , are sequences of nonnegative real numbers in and , , , and the sequence is a small perturbation such that (i), , , and ;(ii), , and ;(iii);(iv), , , and .Then the sequence converges strongly to an element and also is the unique solution of the variational inequality

Proof. By using Lemmas 3 and 5, the mappings and are well defined. First we prove that is bounded. Take , because , are nonexpansive; then we have that
For , because and are nonexpansive and is contractive, we have from (25) that
This implies that
Notice condition ; therefore, is bounded. Consequently, we get that , and , are bounded.Next, we show that . We have from (23) that
Let , ; by the definition of mapping , we have that
Putting in (29) and letting in (30), we have that
Adding (31) and (32), we have that
Since are monotone mappings, which implies that
we have that
that is,
Without loss of generality, let be a real number such that , for all ; then we have that
where . Then we have from (37) and (28) that
On the other hand, let , ; we have that
Let in (39) and let in (40); we have that
Adding (41) and (42) and because are pseudocontractive mappings, we have that
Therefore we have
Hence we have that
where .Let . Hence we have that
Then we have from (46), (45), and (38) that
Notice conditions , , and ; we have that
Hence we have from Lemma 4 that
Therefore we have that
Hence we have from (37), (38), and (45) that
In addition, since , , for all , we have from the monotonicity of , the nonexpansivity of , and the convexity of that
So we have that
Since , , we have from (50) that
In a similar way, we have that
Consequently, we have that
Since the sequence is bounded, there exists a subsequence of and such that weakly. And because , weakly. Next we show that .Because , by the definition of mapping , we have that
Let , , for all ; we have that
Because are monotone and because , we have that
Consequently we have that
If , by the continuity of , we have that ; that is, and then .Similarly, because , by the definition of mapping , we have that
Let , , for all . Because are pseudocontractive mappings, we have that
Because , so we have that
Consequently we have that
that is,
If , by the continuity of , we have that , for all ; we conclude that ; that is, and then . Consequently .Denote ; then is the unique element that satisfies . From Lemma 1, we have that , for all . If we take , then ; consequently we have that .By using the weakly lower semicontinuity of the norm on , we get that
which implies that
Thus, from Lemma 1, we have that
Putting in (68) and in (69), we get that
Adding (70) and (71) we get that ; that is, ; thus . Furthermore, from (67), we get that the sequence strongly and is the solution of the following variational inequality:
Now we show that is the unique solution of the variational inequality , for all .
Suppose that is another solution of the variational inequality; that is,
Let in (72) and let in (73); we have that
Adding (74) and (75), we have that
Hence
Because , we conclude that ; the uniqueness of the solution is obtained. The proof is complete.

Theorem 7. Let be a nonempty closed convex subset of a uniformly smooth strictly convex real Hilbert space . Let be a finite family of continuous pseudocontractive mappings, let be a finite family of continuous monotone mappings such that , and let be a contraction with a contraction coefficient . and are defined as (21) and (22), respectively. Let be a sequence generated by ,
where , , , , are sequences of nonnegative real numbers in and , , , and the sequence is a small perturbation such that (i), , , and ;(ii), , and ;(iii);(iv), , , and .Then the sequence converges strongly to an element and also is the unique solution of the variational inequality

Proof. Take ; because , are nonexpansive, then we have that
For , because and are nonexpansive and is contractive, we have from (28) that
This implies that
Notice condition ; therefore, is bounded. Consequently, we get that , and , , and are bounded.Next, we show that . Denote ; then we get that
Repeating equations from (29) to (38), we have that
Therefore,
Similar to the rest of the proof of Theorem 6, we obtain the result.

If, in Theorems 6 and 7, we let be a constant mapping, we have the following corollaries.

Corollary 8. Let be a nonempty closed convex subset of a uniformly smooth strictly convex real Hilbert space . Let be a finite family of continuous pseudocontractive mappings, let be a finite family of continuous monotone mappings such that , and let be a constant. and are defined as (21) and (22), respectively. Let be a sequence generated by ,
where , , , are sequences of nonnegative real numbers in and , , , and the sequence is a small perturbation such that (i), , , and ;(ii), , and ;(iii);(iv), , , and .

Then the sequence converges strongly to an element and also is the unique solution of the variational inequality

Corollary 9. Let be a nonempty closed convex subset of a uniformly smooth strictly convex real Hilbert space . Let be a finite family of continuous pseudocontractive mappings, let be a finite family of continuous monotone mappings such that , and let be a constant. and are defined as (21) and (22), respectively. Let be a sequence generated by ,
where , , , , are sequences of nonnegative real numbers in and , , , and the sequence is a small perturbation such that (i), , , and ;(ii), , and ;(iii);(iv), , , and .Then the sequence converges strongly to an element and also is the unique solution of the variational inequality

Remark 10. If , then sequence (23) reduces to
and sequence (78) reduces to

The conclusions of Theorems 6 and 7 are true under the same conditions.

Remark 11. Our theorems extend and unify some of the results that have been proved for these important classes of nonlinear operators. In particular, Theorem 6 extends Theorem 6 of Yao and Shahzad [24] in the sense that our convergence is for the more general class of continuous pseudocontractive and continuous monotone mappings. Theorem 6 also extends Theorem 3.2 of Tang [22] in the sense that our convergence is for the more general algorithm with perturbations.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the Natural Science Foundation Project of Chongqing (CSTC, 2012jjA00039) and the Science and Technology Research Project of Chongqing Municipal Education Commission (KJ 1400614).